University of Bucharest Faculty of Mathematics and Computer Science

Conditionals

In natural languages there are lexical entries like if (then), suppose or assume that introduce a conditional relation between a suppositional part and a hypothetically asserted part (usually called the antecedent and the consequent). At the sentence level, Barker and Shan (2008) give the following lexical entry for if:

At the discourse level, as opposed to negation, the scope of the consequent may extend over the sentence boundaries. For example, in “If John owns a book on semantics, he uses it. He also recommends it”, the consequent stretches over the second sentence. The general mechanism of how a language user perceives the logical and rhetorical structure of a discourse is still poorly understood and very far from being solved. Nevertheless, it is certain that a language user knows exactly where the scope of a consequent begins and where it ends. In order to model this kind of conditionals with no a priori delimited scope of the consequent, we give the following lexical entry for if:

The same strategy may be applied to account for cases where the antecedent stretches over several sentences, in discourses such as: Suppose….And suppose….Then….

6. Quantifiers

The puzzling scope behaviour of quantificational determiners is far from being settled, in spite of the abundant research in the area. The most well-known and largely accepted view of natural language quantification is the generalized quantifiers view (Montague 1970, Barwise and Cooper 1981, Larson and Segal 1995, Keenan 2002, Martin van den Berg 1996, Brasoveanu 2007). The generalized quantifier type <<,t>,t> is exactly the type of quantificational determiners in continuation-based semantics. This is by no means a coincidence, generalized quantifiers approach only continuizes the noun phrase meanings rather than continuizing uniformly throughout the grammar as it is done in continuation-based semantics.

A tradition going back at least to Evans (1977) and May (1977) says that the scope of all quantifiers is clause bounded. An E-type (or donkey) pronoun is a pronoun that lies outside the restrictor of a quantifier or outside the antecedent of a conditional, yet co-varies with some quantificational element inside it, usually an indefinite. Here there are some of the famous donkey sentences examples:

“If a farmer owns a donkey, he beats it.”

“Every farmer who owns a donkey beats it.”

Evans (1977) made it standard to assume that the indefinite a donkey cannot take scope over the pronoun it, and therefore cannot bind it, at least not in the ordinary sense of binding. To the contrary, as Barker and Shan (2008) put forward, the relationship between a donkey and it in the above examples seems like binding because it is just binding. In what follows, we adopt the idea that the binding can only occur inside the scope of the binder (all binding is in-scope binding). We account for a wide range of linguistic data involving quantification and anaphora binding.

1. Singular quantifiers

We use the term singular quantificational determiners to refer to quantificational determiners that take as arguments singular common nouns, such as a, every, any, each or no. We will not commit ourselves to the hypothesis that the scope of all quantifiers is clause bounded. The important assumption here is that the binding can only occur inside the scope of the binder (all binding is in-scope binding). Indefinites are notorious for their binding capabilities outside their minimal clause, as opposed to the quantificational determiners every, any or no. Thus, one may say that the quantificational determiner a (that combines in English with a common noun to form an indefinite) can take scope outside its minimal clause, whereas every, any and no cannot.

Shan & Barker (2008) propose the following lexical entry for every:

Here, the authors apply the idea of distinguishing between the entire scope of a quantifier, its restrictor scope and its nuclear scope. The gap in the top layer of the semantic component of every indicates the entire scope of the quantifier. The variable P is the restrictor and the negated gap delimits the nuclear scope.

As Barker and Shan (2008) rightfully note, there is no obvious semantic reason why every cannot extend its scope outside its minimal clause. They presume it must be a syntactic reason. We assume there are two different meanings of every, exemplified by the two examples:

“Every man came. *He whistled” (distributional meaning)

“Every man loves a woman. He gives her flowers to prove it to her” (generic, universal meaning)

The use of every from the first example means “all men came”, whereas the second meaning is generic and may be paraphrased as “for any way of choosing a particular man,…”. We leave implicit here the mechanism for deciding which meaning is intended in a particular case.

The stipulation that every is polymorphic (it has two different meanings) is supported cross-linguistically by the fact that in the two examples above, every translates in Romanian by two different quantificational determiners: the distributional ‘fiecare’ and the generic ‘oricare’, respectively and not vice-versa.

The use of logically equivalent form in terms of existential quantifier and negation is just a matter of convenience (to have only existential quantification and negation in logical forms); we could have just as well used the lexical entry for every in explicit terms of universal quantification like:

The generic use of every may be modeled by choice functions; we propose the following lexical entry for the generic use of every:

where f is a function that takes a property and returns a constant (an individual), thus of type <,e>. It is necessary to consider only those functions that map each (non-empty) set to a member of that set. This and other technical details can be found in (Egli 1995). As we have assumed that a binder can bind an anaphoric expression only inside its scope, it is natural to say that in such cases the nuclear scope of choice-function every extends to the following sentence. The singular pronoun refers back to the particular man f(man), thus a constant. Here is the interpretation of the upper example:


The lexical entry for plural common nouns will be introduced in the Plural Quantifiers section. So, for now, we do not pause to explain the semantics of flowers used here.






This interpretation is intuitive enough, because a paraphrase for the upper discourse is “for every way of choosing a man there is a woman this man loves and there are some flowers this man gives to the woman”.

We can give the related quantificational determiner any a similar lexical entry; the interpretative difference between every and any is made (in line with Quine and Geach among others) by the scope behavior of the two quantificational determiners. Any prefers to take wide scope, whereas every prefers to take narrow scope. The most common use of any is in negative sentences (any is a negative polarity item) and questions. Examine for instance the two sentences:

“John does not know every poem.”

“John does not know any poem.”

The difference in meaning between the two sentences is given by the scope preferences of the two quantificational determiners.




which means that there is at least one poem that John does not know, a fare approximation of the intended meaning.



which means that there is no poem that John knows, also a fare approximation of the intended meaning.

When used in negative context, due to the fact that any semantically implies nonexistence, it cannot take scope outside its minimal clause, exactly like not and no. For instance, consider: “John does not know any poem. *It is nice.”

We have to ensure the scope closing in the usual way, by applying Lower Rule immediately after the interpretation of its minimal clause. It cannot be argued that it is the negation which prevents further referring to any poem, because any takes wide scope over negation.

Notice that there is a third intermediate possibility of scope taking, with negation taking scope at the second level of the compositional tower:

The quantificational determiner each is very similar to every. There has been said that each is “more distributional” than every. Specifically, all the occurrences of each may be replaced by occurrences of every, without losing the grammaticality, but not the other way around. Every requires some distributivity, while each requires a total form of distributivity (where there is a significant individual difference between the objects over which each distributes). Arguably, it seems that events or situations or possible worlds are involved in distinguishing the semantics of every and each.

Another interesting quantificational determiner is no, for which we propose the following lexical entry:

To exemplify, the sentence “No man came” has the following derivation:


As in the case of not, we have to force the closing of no’s minimal clause, by applying Lower immediately after the interpretation of that clause, in order to disallow subsequent anaphoric reference as in “No man came. *He whistled”.

It is worth mentioning that when a quantificational determiner such as a, every, any or no appears in a DP having direct object position, no type mismatch occurs and the quantificational determiner is smoothly interpreted in situ without any further stipulations, as opposed to other theories in which such a type mismatch has to be fixed (for example by quantifier raising rule in QR theory). For instance, here is the derivation of “John saw no man”:

2. Plural quantifiers

There is a vast literature on representing plurals. We will only refer to four of the most influential existing approaches: the proposals of Scha (1981), Link (1983), Roberts (1987) and of Verkuyl and van der Does (1991).

Plural quantificational determiners take as arguments plural common nouns. Plural quantificational determiners (among other constructions such as coordinated DPs or singular DPs), introduce plural referring variables. From a technical point of view, a plural referring variable notated with upper letters (X, Y, Z,..) is a set of entities.

We will give lexical entries for the plural quantificational determiners some (not to be confused with its singular counterpart), all and most, in the continuation semantics framework. Some and most need special care when the plural variable they introduce binds some subsequent anaphora, due to the so-called maximality constraint. While Some kids came (with no other subsequent anaphor that refers to the kids that came) means that there is a set of any cardinal of kids that came, the discourse “Some kids came. They played” means that there is a maximal set of kids who came and that maximal set played. So, there is a maximality operator that blocks further transmission of arbitrary sets, much like the negation blocks transmission of values of indefinites in direct object position to subsequent anaphora. The two uses of some have different truth-conditions. When some is used in the first, weak sense, we take it to have the following lexical entry:

When used in the second, maximal sense, that exports a maximal set to bind a subsequent anaphora (such as they), we take some to have the alternative lexical entry:

For the same reasons, we similarly treat the quantificational determiner most, for which we propose the following two alternative lexical entries, one for the weak sense, and one for binding sense, respectively:

We turn now to the problem of compositionally obtaining the meaning of bare plurals. Bare plurals are plurals without overt article. Sentences with bare plurals can have existential (“John gave Mary flowers”) and universal or generic (“Flowers are beautiful”) readings. We propose that the existential reading is accounted for in this framework by a silent quantificational determiner that has the same semantics as some (i.e. the two weak and maximal senses). The universal reading is accounted for by a similar silent quantificational determiner, having the semantics of all:

We take predicate to be distributive or collective (see Distributivity chapter), so they are responsible for deriving the right truth-conditions (in “John gave Mary flowers”, gave is used in its collective sense for its second argument; in “Flowers are beautiful”, is beautiful is used in its distributive sense in its first argument.)

Cardinal determiners have two built-in meaning components: an existential component and a cardinality one. We propose the following two alternative lexical entries for card, one for the meaning there are card Ps…, the other for the meaning there are exactly card Ps…:

The semantic ambiguity between the two lexical entries of a cardinal card is determined by whether the scope of the following context (continuation) lies inside the scope of the cardinality (as in the second entry) or not (as in the first entry).

Both these (weak and strong) meanings of the cardinals may bind subsequent anaphora (as opposed to the case of some that can bind only with its maximal meaning). For the weak meaning, we can just use the regular Bind rule, whereas for the strong meaning (exactly card), one cannot use Bind because that would bring the continuation into the scope of argmax, altering the truth conditions. Thus, we have to force the scope closing of argmax immediately after the interpretation of the cardinal’s minimal clause by applying Lower. To allow the strong meaning of card to bind, we have to give it jet another lexical entry:

1. Distributive vs. collective readings

We will consider two of the most influential existing strategies to deal with plurals and their associated ambiguities (collective, distributive or cumulative readings): Scha (1981) and Link (1983, 1991). Scha and Link locate the source for the ambiguity of plural sentences differently. According to Scha the ambiguity between collective, distributive and possibly other readings is located in the plural noun phrase or more precisely in the determiner. According to Link, noun phrases are unambiguous and the readings should be generated within the verb phrase. A third strategy proposes that readings of complex sentences are a result of the whole structure or as Roberts (1987, p. 100) puts it: “Distributivity is a property of predications, combinations of a subject and a predicate.”The readings can be triggered by different elements of a sentence; there is a functional interplay between the different categories.”

We will take predicates, not nouns to be distributive, collective, or ambiguous. We will not commit ourselves to whether the distributivity comes as a feature from the lexical semantics, or it is entailed from the world knowledge and the sense of the predicate itself (Roberts 1987).

Here are some examples:

“Sue and Mary are pregnant.” (be pregnant is a distributive predicate)

“John and Bill moved the piano.” (moved is an ambiguous (between distributive and collective) predicate)

“The students gathered in the square. “ (gathered is a collective predicate)

As a general rule, we posit that a distributive predicate P_dist is true of a plural referring variable X={x₁, x₂, …x_n} iff P_dist(x₁) P_dist (x₂) P_dist(x_n). And a collective predicate P_call is true of a plural referring variable X={x₁, x₂, …x_n} iff P_call().Note that a predicate may have multiple arguments (subject and direct object, for instance). So a predicate may be distributive or collective in each of the arguments.

2. Coordination: conjunction and disjunction

Barker (2002) proposes the following lexical entry for or:

We point that disjunction may introduce only singular variables, as we can see from the following examples:

“John owns a donkey or a goat. He beats it/* them.”

“John or Bill called. He/*They hang up.”

We straightforwardly extend Shan and Barker’s (2008) semantic representation for disjunction to conjunction:


Note that and is also polymorphic. Thus it may account for discourses like: “John drinks and talks. He does this for hours” where this is anaphoric to plural events, provided only we modify the binding rule to allow categories other then DP (like DP\S) to bind subsequent pronouns. Note also that conjoined DPs have the power to introduce variables that may be further referred by plural pronouns, a power disjoint DPs do not have. To illustrate this, consider the following examples:

“John owns a donkey and a goat. He beats *it/ them.”

“John and Bill called. *He/They hang up.”

3. Plural pronominal anaphora

Ideally, singular and plural pronominal anaphora should behave similarly and be parallel phenomena. Unfortunately, at a closer look, there are striking differences between the anaphoric properties of singular and plural pronouns. On the one hand, only singular DPs have the power to introduce singular variables that could bind subsequent singular pronominal anaphora. On the other, plural variables may be introduced not only by plural DPs, but also by:

• 
  Two or more singular DPs, coordinated (“John and Mary came. They whistled”) or not (“John took Mary to Acapulco. They had a lousy time”),
  
• 
  Quantificational singular DPs (“Every man came. They whistled” or “A kid climbed every tree. They were full of energy”).
  

For “Most kids came. They played”, the derivation is:

The derivation for “All kids came. They played” is:

As for the plural anaphora introduced by cardinal determiners, consider the following two examples:

“Five men walk in the park. They watch the birds” (preferred reading: there are some context relevant five men and they walk in the park and they watch the birds; there could be other not contextually important men walking and watching);

“Five men walk in the park and watch the birds” (preferred reading: there are exactly five men who are in the park and watch the birds).

Evans (1977) gave this sort of examples and others (such as the donkey-sentences) to assert that there are two types of pronouns: bound-pronouns (as in the second example) and E-type pronouns (as in the first one). He assumes that quantifiers cannot extend their scope over clause boundaries. We already argued that this is not the case (in line with Barker and Shan (2008)) at least for indefinites. Cardinals are also good examples of quantificational determiners that may extend their scope over their minimal clause limits. For the above examples, what happens is that both are semantically ambiguous between two readings which correspond to the two scope-distinct lexical entries for the cardinal determiner five. Pragmatic reasons dictate the preferred reading in each case. We give the derivations of these preferred readings (and skip the not preferred ones, though semantically possible), ignoring the full interpretation of walk in the park and of watch the birds:




=
which means that is a set of cardinality five composed of men who walk in the park and watch the birds.






which means there are exactly five men in the park who watch the birds.

We turn now to the case of introducing plural entities by coordination (conjunction or disjunction). The lexical entry for conjunction obviously gives right truth conditions and offers an antecedent for subsequent anaphora, as in, for example: “John and Mary came. They whistled”. In such cases, where more than one specific antecedent is present in the discourse, the lexical entry for they needs to search left for two (or three, or another number) DPs, for instance:

Note that reversing the order of Binding and Lifting for the two DPs gives the denotation. As usual, the layers act like indices; a superior level takes scope at inferior levels and left expressions take scope at right expressions, to account for left-to-right natural language order of processing.

An open problem still remains: how to block “John or Bill called. *They hang up”?

The third case, the case of introducing plural entities by singular DPs is the most difficult. We will stipulate that a singular DP may bind a plural entity (introduced by a pronoun or a definite) if and only if it is logically a plural, that is:

1. 
   Either the singular DP is bound by universal quantifier,
   
2. 
   Or the singular DP is embedded inside an expression in which it co-varies with a variable bound by the universal quantifier (the so-called structural dependency).
   

“A kid climbed every tree.” followed by

“He was full of energy.” or

“They were full of energy.”

Note that the first sentence has two distinct readings, one in which a takes scope over every and one in which every takes scope over a. If the first sentence is continued by the second, then the only possible reading in natural language becomes that with a taking scope over every. If the first sentence is continued by the third, then the only possible reading in natural language becomes that with every taking scope over a. Ignoring the tense, for simplicity, the interpretations where every takes narrow scope is:








Note that it is impossible that a plural pronoun (such as they) be bound by a kid in this interpretation, because the variable x is bound simply by the existential quantifier.

When every takes wide scope, it does so both with its general scope and its nuclear scope:

Note that, similar to the case of wide-scope any, there is a third intermediate scope-taking possibility, which gives an impossible interpretation for natural language and supports the hypothesis that every takes wide scope obligatory with both general and nuclear scope:

“No man came. They were ill”

It may be said that they refers to the noun man and not to the DP no man, or that they refers to the complement set of the DP no man. Either way, no suitable lexicalized antecedent is available in the discourse for they to refer back to. It could be argued that no (and other quantifiers) introduces both their variable (set of variable) and the complement set (w.r.t some context given set). We will have no more to say about these cases here.

3. Conclusions

We showed how singular and plural quantifiers can be represented in continuation semantics framework. We discussed the scope behaviour of singular quantifiers. We accounted for some aspects of plural dynamic semantics such as plural anaphora, conjunction and disjunction, distibutivity or the maximality condition.

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